Hypersymplectic Structure

A hypersymplectic structure on an oriented four-dimensional smooth manifold is a triple of closed real 2-forms whose span at each point is a positive three-dimensional subspace for the wedge-product pairing. More explicitly, for any positive volume form , the matrix defined by is a positive definite matrix. Every nonzero constant linear combination of the three forms is symplectic. The triple determines a conformal Riemannian structure and need not initially be parallel for its induced metric. A hyper-Kähler triple supplies an example. The hypersymplectic deformation conjecture concerns deformation to such a parallel triple on a closed manifold.